Sn, C, and Si, Ge are all group XIV elements. Yet, Sn is a conductor, C is an insulator while Si and Ge are se

NCERT Exemplar · Q13

Q. Sn, C, and Si, Ge are all group XIV elements. Yet, Sn is a conductor, C is an insulator while Si and Ge are semiconductors. Why?

Uttar Pradesh UpmspShort· 2mImportance★★★★★

73% · 27/37 Questions

🔒 Locked · start free trial →

You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.

Start your 14-day free trial to unlock the full solution →

Concept understanding — Band Gap Energy

What is Band Gap Energy? — A First Look

Imagine you have a single atom. Its electrons live in specific, fixed energy levels — like rungs on a ladder. You can't put an electron halfway between two rungs; it's either on one rung or another.

Now bring two atoms close together. Their electron rungs interact and split into two slightly different energies. Bring a billion atoms together — as in a solid crystal — and those original rungs spread into continuous bands of allowed energies, separated by gaps where no electron can exist.

That gap — the forbidden region between two bands — is the band gap.

The Intuition: A Wall Between Two Rooms

Think of the valence band as the ground floor of a building — electrons here are tightly bound to atoms, not free to move. The conduction band is the first floor above — electrons here can roam freely through the crystal, carrying current.

The band gap is the height of the ceiling between these two floors. An electron needs exactly that much energy to jump from the valence band to the conduction band. If you give it less energy, it stays stuck on the ground floor. If you give it exactly the gap energy or more, it can leap up and become a mobile charge carrier.

Note

In a metal, the valence and conduction bands overlap — there is no gap. That's why metals conduct electricity so easily: electrons already have free states available at no energy cost.

The Precise Statement

Band gap energy ( $E_g$ ) is the minimum energy required to excite an electron from the top of the valence band to the bottom of the conduction band in a solid.

E_g = E_{\text{conduction band minimum}} - E_{\text{valence band maximum}}

It is measured in electron volts (eV). One eV is the energy gained by an electron when accelerated through a potential difference of 1 volt — a tiny but convenient unit for atomic-scale energies.

Why Does It Matter?

The band gap determines almost everything about how a material behaves electrically and optically:

Material type	Typical $E_g$	Behaviour
Conductor (metal)	$E_g = 0$ (bands overlap)	Electrons flow freely at room temperature
Semiconductor	$0.1 \text{ eV} < E_g < 3 \text{ eV}$	Conducts only when given energy (heat, light)
Insulator	$E_g > 3 \text{ eV}$	Almost no conduction at normal conditions

Tip

A quick rule of thumb: if a material is transparent to visible light, its band gap is larger than about 3.1 eV (the energy of violet light). Diamond ( $E_g \approx 5.5$ eV) is transparent; silicon ( $E_g \approx 1.1$ eV) is opaque and shiny.

A Concrete Example: Silicon

Silicon has a band gap of 1.12 eV at room temperature. This means:

An electron in the valence band needs at least 1.12 eV to jump to the conduction band.
Visible light photons have energies between 1.8 eV (red) and 3.1 eV (violet). So silicon absorbs most visible light — that's why solar cells are dark. …

Why this formula?

Band Gap Energy: Why the Formula Holds

The band gap energy $E_g$ is the energy difference between the top of the valence band and the bottom of the conduction band in a solid. The key formula is:

E_g = E_c - E_v

where $E_c$ is the minimum energy of the conduction band and $E_v$ is the maximum energy of the valence band.

But why does this simple difference matter? The answer lies in how electrons behave in a crystal.

The Origin of Energy Bands

In an isolated atom, electrons occupy discrete energy levels. When atoms come together to form a solid, their atomic orbitals overlap. According to the Pauli exclusion principle, no two electrons can occupy the same quantum state. So the discrete levels split into a continuum of closely spaced levels — an energy band.

The valence band is formed from the outermost (valence) atomic orbitals. The conduction band is formed from the next higher set of orbitals (typically the empty orbitals above the valence orbitals). Between these bands lies the band gap — a region of forbidden energies where no electron states exist.

Why the Formula $E_g = E_c - E_v$ Is Not Trivial

You might think: "Of course the gap is the difference between the bottom of one band and the top of another." But the real insight is that $E_c$ and $E_v$ are not arbitrary points — they are the extrema of the band structure.

In a periodic crystal, the electron energy $E(\mathbf{k})$ depends on the wavevector $\mathbf{k}$ . The valence band has its maximum at some $\mathbf{k}$ -point (often at $\mathbf{k}=0$ for direct-gap semiconductors), and the conduction band has its minimum at some $\mathbf{k}$ -point. The band gap is:

E_g = \min_{\mathbf{k}_c} E_c(\mathbf{k}_c) - \max_{\mathbf{k}_v} E_v(\mathbf{k}_v)

This is not just a difference — it's a minimisation over all possible electron momenta.

Why This Difference Determines Conductivity

The band gap controls whether a material is an insulator, semiconductor, or conductor because of the Fermi-Dirac distribution:

f(E) = \frac{1}{1 + e^{(E - E_F)/k_B T}}

At absolute zero, all states below the Fermi level $E_F$ are filled, and all above are empty. For an intrinsic semiconductor, $E_F$ lies in the middle of the band gap. The probability that an electron is thermally excited from the valence band to the conduction band is proportional to $e^{-E_g/2k_B T}$ .

Important

The band gap energy $E_g$ appears in the exponent of the carrier concentration formula:

$n = p = N_c N_v e^{-E_g/2k_B T}$

This is why a small change in $E_g$ causes a huge change in conductivity — it's an exponential dependence.

The Physical Meaning of $E_g$

The band gap is not just a number — it's the minimum energy required to:

Break a covalent bond in the crystal (creating an electron-hole pair)
Promote an electron from a bonding state to an antibonding state
Create a mobile charge carrier

For example, in silicon ( $E_g = 1.12$ eV at 300 K), a photon with energy greater than 1.12 eV can be absorbed, exciting an electron from the valence band to the conduction band. This is why silicon is used in solar cells — the band gap matches the solar spectrum. …

Start your 14-day free trial to unlock the full solution →

The electrical behaviour of group XIV elements (C, Si, Ge, Sn) is determined by the size of their band gap — the energy difference between the valence and conduction bands. Diamond (C) has a very large band gap ( $\approx 5.5\ \text{eV}$ ), making it an insulator. Si ( $\approx 1.1\ \text{eV}$ ) and Ge ( $\approx 0.7\ \text{eV}$ ) have moderate band gaps, making them semiconductors. Sn (grey tin) has a nearly zero or overlapping band gap, behaving as a conductor.

The key lies in how the band gap energy changes as you go down group XIV. Let’s understand why.

The Concept: Band Gap and Bonding

In a solid, atomic orbitals overlap to form energy bands. The valence band is filled with electrons; the conduction band is empty. The band gap ( $E_g$ ) is the energy an electron needs to jump from the valence band to the conduction band — to become free to conduct electricity.

For group XIV elements, each atom has four valence electrons. In the solid, these form covalent bonds. The strength of these bonds determines how wide the band gap is.

Bond strength and atomic size: As you go down the group (C → Si → Ge → Sn), atomic size increases. Larger atoms have valence electrons that are farther from the nucleus and more loosely held. This means the covalent bonds between atoms become weaker.
Weaker bonds → smaller band gap: A weaker bond means the energy required to break an electron free (to promote it to the conduction band) is smaller. So, the band gap decreases down the group.
The critical threshold: For electrical conduction, thermal energy at room temperature ( $k_B T \approx 0.025\ \text{eV}$ ) can promote a small number of electrons if the band gap is small enough (a few eV). If the band gap is huge, almost no electrons can jump — the material is an insulator. If the band gap is zero or negative (bands overlap), electrons flow freely — the material is a conductor.

E_g(\text{C}) \approx 5.5\ \text{eV},\quad E_g(\text{Si}) \approx 1.1\ \text{eV},\quad E_g(\text{Ge}) \approx 0.7\ \text{eV},\quad E_g(\text{Sn}) \approx 0\ \text{eV}

Now, let’s apply this to each element.

Step-by-Step Reasoning

Carbon (Diamond): Carbon is the smallest atom in the group. Its four valence electrons form extremely strong, directional covalent bonds (the diamond structure). This gives a very large band gap of about $5.5\ \text{eV}$ . At room temperature, thermal energy is far too small to excite electrons across this gap. Hence, diamond has almost no free electrons and is an insulator. …

Start your 14-day free trial to unlock the full solution →

←Previous questionQ12 — Why are elemental dopants for Silicon or Germanium usually chosen from…→Next questionQ14 — Can the potential barrier across a p-n junction be measured by simply …

Q.Sn, C, and Si, Ge are all group XIV elements. Yet, Sn is a conductor, C is an insulator while Si and Ge are semiconductors. Why?

Concept understanding — Band Gap Energy

What is Band Gap Energy? — A First Look

The Intuition: A Wall Between Two Rooms

The Precise Statement

Why Does It Matter?

A Concrete Example: Silicon

Why this formula?

Band Gap Energy: Why the Formula Holds

The Origin of Energy Bands

Why the Formula $E_g = E_c - E_v$ Is Not Trivial

Why This Difference Determines Conductivity

The Physical Meaning of $E_g$

The Concept: Band Gap and Bonding

Step-by-Step Reasoning

Method: Tracking How Bond Strength (and Hence Band Gap) Changes Down a Group

Q.Sn, C, and Si, Ge are all group XIV elements. Yet, Sn is a conductor, C is an insulator while Si and Ge are semiconductors. Why?

The Concept: Band Gap and Bonding

Step-by-Step Reasoning

🎓Unlock everything free for 14 days

Unlock everything free for 14 days